Transcript 2 Better Maths Teaching at KS2

Principles of Good Maths Teaching
Objectives:
•Understand the essence of ‘Good Maths Teaching’
•Look at some activities that encourage ‘Good Maths Teaching’
•Look at important skills necessary to make progress after KS2:
Specifically look at resources and issues around
a) Fractions
b) Inverse Operations
c) Proportional Reasoning
Maths Activity A – 5 minutes
1
2
3
4
5
6
7
8
9
10
Number counters 1 to 10.
Use pairs to make 10
Which numbers are left over? Why?
What if you had to make a total of 11? 12? 13? 14?
Principle 1:
Pupils understand maths when they are using and doing maths.
The trick is to set up situations where the pupils can use maths in
open ended ways.
Fewer closed questions and more open questions that let them
explore their own reasoning.
Good maths teaching is about developing understanding
Maths Activity B – 5 minutes
Either:
Write down 3 different numbers that sum to be 12?
How many ways can you do this?
Or:
Twelve can be written as the sum of 3 consecutive numbers. What are
they?
Can you find 5 more numbers that can be written as the sum of 3
consecutive numbers?
Can you find 5 numbers that can be written as the sum of 5 consecutive
numbers?
Can you find 5 numbers that can be written as the sum of 5 consecutive
numbers and 3 consecutive numbers?
Principle 2:
Start at 10 and count back to -10
How many numbers did you say?
Principle 2:
Start at 10 and count back to -10
How many numbers did you say?
We use numbers for two purposes:
1) To count
2) To measure
Often we spend too much time focusing on counting rather than
measuring.
Measuring is more intuitive and more kinaesthetic/visual.
It is an underused tool in developing number sense.
Maths Activity C – 5 minutes
0
10
20
30
40
50
0
1
2
3
4
5
What is 20 ÷ 10? What is 2 ÷ 1? What is 10 ÷ 5? What is 1 ÷ 0.5?
How many divisions can you write down where the answer is 2?
Principle 3:
The Numeracy Strategy had a focus on mental methods. Excelling at
mental maths requires many different tricks to get to the answers.
BUT too many techniques can cause confusion with weaker students
It is better to agree on a standard school method for adding,
subtracting, multiplying and dividing and work towards
developing it without getting distracted by other methods.
Use ‘clever tricks’ or new methods only when pupils are secure in the
main method.
Maths Activity D – 5 minutes
Blonde
Hair
Only 1 face.
How many
ways?
What about 2?
or 3?
Most?
Sad
Making Rich Maths Tasks:
Turning a Closed Question into an Open Question
Closed:
Open:
Find the perimeter of these
shapes.
Can you find 6 items in the
room with a Perimeter of
between 20 cm and 50 cm?
Closed:
What is 14 divided by 6?
Open:
Closed:
Open:
What is 14 divided by 6?
I think of a number. When I
divide it by 6 I get a remainder
of 2. What number might I have
thought of?
When I divide my number by 5 I
get a remainder of 3.
What number might I have
thought of?
Closed:
Find the perimeter of these
shapes/items.
Open:
Closed:
Open:
Find the perimeter of these
shapes/items.
Suppose you have 5 cm squares.
How many different shapes can
you make from them.
Which shape has the largest
perimeter?
Which has the smallest
perimeter?
OR
If you had 12 cm squares
Find the shape(s) with the
largest/smallest perimeter?
Are you are answers unique?
How do you know?
Closed:
What is the next number in this
sequence:
2, 4, 6, 8, 10, ……
Open:
Closed:
Open:
What is the next number in this
sequence:
What is the next number in this
sequence:
1, 3, ……
2, 4, 6, 8, 10, ……
OR
Write down 4 numbers in a
sequence
OR
Here are some numbers.
Give a reason why each of them
might be the odd one out:
6, 15, 30, 40
Closed:
Open:
What is the next number in this
sequence:
Hand out a variety of 2d shapes –
different colours, sizes, number of
sides
2, 4, 6, 8, 10, ……
Make a path with according to some
rule: Eg: red, blue, yellow, red, blue
yellow, red blue
yellow,
Or 3 sides 4, sides 3, sides, 4
sides
Or Triangle, square, triangle
square,
Or Triangle, square,
pentagon,
hexagon,
Describe your pattern to someone
else? Can they add another shape in
Closed:
Construct a bar chart
for 3 red cars and 5 blue cars
Open:
Closed:
Open:
Construct a bar chart
for 3 red cars and 5 blue cars
What is this a graph of:
Closed:
What is the product of 8 and
16?
Open:
Closed:
Open:
What is the product of 8 and
16?
The product of two whole
numbers is 128.
What were the two numbers?
How many possibilities?
Closed:
Open:
Find the missing number:
8+
= 15
Closed:
Open:
Find the missing number:
How many pairs of numbers can
you find that sum to 15
8+
= 15
OR
+
= 15
Closed:
What shape is this?
Open:
Closed:
Open:
What shape is this?
Draw as many different shapes
as you can in the next 5 minutes
Or
Hand out a loop of string – what
shapes can you make?
Get pupils into pairs what
shapes can they make?
Suppose you had groups of 3
and each child pulled on the
string to make it taut – what
shape would you make? Would
every group make the same
shape?
What if you had groups of 4?
Maths Activity E – 5 minutes
I arrange the pupils in KS2 into rows with an equal number
in each row.
When I have rows of 3 I have one pupil left over.
When I have rows of 4 I have one pupil left over.
When I have rows of 5 I have one pupil left over.
When I have rows of 6 I have one pupil
left over.
How many pupils in KS2?
Is there only one answer?
Important Skills Required to Bridge the Gap from KS2 to KS3
1) Being able to add and subtract whole and decimal numbers
Place value is key. Money is a good approach but also use length.
Subtraction 12.8 – 1.43 always an issue
2) Recall of Multiplication tables
Written method to multiply two digit whole numbers together
Grid Method is a good approach that leads to more developed written techniques
3) Division
Chunking leading to short division
You can’t do division unless you know your times tables
Remainders drive me nuts!
Important Skills Required to Bridge the Gap from KS2 to KS3
4) Inverse Relationships
That 5 + 6 = 11 is equivalent to 11 – 6 = 5
11 – 5 = 6
That 4 × 3 = 12 is equivalent to 12 ÷ 3 = 4
12 ÷ 4 = 3
5) Understanding what a fraction really represents
Understand that 2/5 means two out of 5 equal parts for example
Understand that the denominator is a label representing the number of equal
parts in the whole.
6) Proportional Reasoning
If 6 pints of mile costs me £2.40 how much will 7 pints of milk cost.
Using the unitary method.
Resources for helping with Number Work:
Put the numbers 1 to 6 into the circles so that the circle resting on
each pair is equal to the difference in their values
Repeat with 10 circles and the numbers 1 to 10
Pick any three numbers between 1 and 9 to go in the bottom three
circles. Each circle’s value is the product of the two numbers it is
resting on.
What is the biggest total you can make in the top circle?
What is the smallest total? What is the largest odd total?
Write down 3 consecutive numbers:
Square the middle number and multiply the outside two.
What is the difference in your answers?
Repeat with a different set of 3 numbers …….
Write down 4 consecutive numbers. Find the difference in
the product of the 1st and 4th and the 2nd and 3rd?
What do you notice?
Repeat with 5 numbers? 6 numbers?
Strike it Out
You need a friend to play with
One of you draw a 0-20 number line like this
3 + 8 = 11
11 + 9 = 20
20 - 4 = 16
The answers to these three questions use 8 of the 9 digits in the grid.
Which digit is not used?
13 x 19 = ?
4 5
8
14 x 7 = ?
1 2
1
119 + 32 = ?
7 9
1
Make up your own problem? Design your own grid.
Can you find three questions to match a grid that has 9
different digits in it!
Resources for helping with Inverse Relationships:
Ergo Grids
Using length/measurement:
12 cm
Using these diagrams:
5 cm
2 cm
Clearly 12 – 5 – 5 = 2 is the same as 5 + 5 + 2 = 12
Or
You can use this to solve 5 + 5 +
2×
= 12
+ 2 = 12
Balancing Scales from Nrich:
Suppose you have a number machine that always does the same
thing. It is first set up to add 10.
I put three numbers into the machine:
The first number gives me 12
The second number gives me 15
The third number gives me 8
What numbers did I put in the machine.
I then reset the machine so that it is doing a different add or take
away and I put in 4 new numbers. One of the numbers that I put
in the machine is the number 10.
If the outputs are:
0 , 19, 1 and 11
What numbers did I put in the machine?
Resources for addressing issues with fractions
Pupils struggle to understand that the denominator
doesn’t represent an amount but is really a label telling you how
many parts the whole has been chopped into
We often get pupils to colour in diagrams like:
this to represent 3/4 . It is better to get them
to do their own partitioning.
What fraction of each
shape is shaded?
Draw a picture to show ½ + 1/4?
Fractions need to used to describe as wide a range of situations as
possible:
Discrete wholes:
Eg 3 red marbles and 5 blue
Continuous wholes: Eg Cut this circle into thirds (measurements!)
Definite wholes:
What fraction of
shapes are hearts?
Indefinite wholes: Every Monday since I started work I have bought
myself a kit-kat at lunch time. What fraction of
days have I had a kit-kat for lunch?
It is easy to get stuck on halving and (quartering halving a half…)
Make certain that pupils are exposed to other fractions at an early
age (thirds, fifths etc)
Encourage the children to make meaningful comparisons
E.g. ‘I invite 4 children to a part and order 3 pizzas’ Does every
child get more or less than half a pizza? How much does each child
receive?
‘What if I invited 8 children and ordered 6 pizzas’ would each child
get more or less pizza?
‘If I invite 5 children and order 4 pizzas’ will each child get more or
less pizza?
= 1 whole?
a)
What fraction is
b)
What fraction is
?
?
Resources for helping with Proportional Reasoning:
Eg. If 6 pints of mile costs me £2.40 how much will 7 pints of milk cost.
Using the unitary method with a diagram.
40 p
40 p
40 p
40 p
40 p
40 p
To make 20 litres of orange paint you need 8 litres of red paint
and 12 litres of yellow paint
Red
8
Yellow
12
Total
20
10
40
16
60
200
9
18
75
Maths Activity F – 5 minutes
You have a blank 100 square
Where do these numbers go on this number grid?
1, 5, 13, 24, 45, 30, 75, 99
What if the grid was a 9 by 9 square?
A hundred square has been printed on both sides of a piece of
paper. One square is directly behind the other.
What is on the back of 100? 58? 23? 19?
Can you see a pattern?
Maths Activity G – 5 minutes